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Construction of the Outer Automorphism of Via a Complex Math.Comput.Sci. (2018) 12:453–458 https://doi.org/10.1007/s11786-018-0382-0 Mathematics in Computer Science Construction of the Outer Automorphism of S6 via a Complex Hadamard Matrix Neil I. Gillespie · Padraig Ó Catháin · Cheryl E. Praeger Received: 15 December 2017 / Revised: 20 March 2018 / Accepted: 22 March 2018 / Published online: 26 September 2018 © The Author(s) 2018 Abstract We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers. Keywords Complex hadamard matrix · Outer automorphism · Symmetric group Mathematics Subject Classification Primary 20B25; Secondary 05B20 · 20B30 1 Introduction Sylvester showed that the fifteen two-subsets of a six element set can be formed into 5 parallel classes in six different ways and that the action of S6 on these synthematic totals is essentially different from its natural action on six points [13]. To our knowledge this was the first construction for the outer automorphism of S6. Miller attributes the result that for n = 6, Sn has no outer automorphisms to Hölder, and Sylvester’s construction of the outer automorphism of S6 to Burnside [11]. He also gives a by-hand construction of the outer automorphism. The papers of Janusz and Rotman, and of Ward provide easily readable accounts which are similar to Sylvester’s [10,14]. Cameron and van Lint devoted an entire chapter (their sixth!) to the outer automorphism of S6 [2]. They build on Sylvester’s construction to construct the 5-(12, 6, 1) Witt design, the projective plane of order 4, and the Hoffman–Singleton graph. Via consideration of the cube in R3, Fournelle gives a heuristic for the existence of an outer automorphism of S6, and constructs it with the aid of a computer [7]. Howard, Millson, Snowden and Vakil give two constructions of N. I. Gillespie (B) Heilbronn Institute for Mathematical Research, University of Bristol, Tyndall Ave, Bristol BS8 1TH, UK e-mail: [email protected] P. Ó Catháin Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA e-mail: [email protected] C. E. Praeger School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia e-mail: [email protected] 454 N. I. Gillespie et al. the outer automorphism of S6, and use this to describe the invariant theory of six points in certain projective spaces [9]. In this note we give a construction which we believe has not previously been described, using a complex Hadamard matrix of order 6 and a representation of the triple cover of A6 over the complex numbers. This note is inspired by a construction of Marshall Hall Jr [8] for the outer automorphism of M12 via a real Hadamard matrix of order 12, and by Moorhouse’s classification of the complex Hadamard matrices with doubly transitive automorphism groups [12]. It was in the latter paper that we first became aware of the complex Hadamard matrix of order 6 discussed in this article, where it is described as corresponding to the distance transitive triple cover of the complete bipartite graph K6,6. 2 Hadamard Matrices Let ω be a primitive complex third root of unity. Then the matrix H6 is complex Hadamard. ⎛ ⎞ 111111 ⎜ ⎟ ⎜ 11 ω ω ωω⎟ ⎜ ⎟ ⎜ 1 ω 1 ω ω ω ⎟ H6 = ⎜ ⎟ ⎜ 1 ωω1 ω ω ⎟ ⎝ 1 ω ωω1 ω ⎠ 1 ω ω ωω1 † = † This means that H6 satisfies the identity H6 H6 6I6, where for an invertible complex matrix A, A is the complex conjugate transpose of A. Equivalently, H6 reaches equality in Hadamard’s determinant bound. We refer the reader to [6] for a comprehensive discussion of Hadamard matrices and their generalisations. An automorphism of a complex Hadamard matrix is a pair of monomial matrices (P, Q) such that P−1 HQ = H. The set of all automorphisms of H forms a group under composition. In this note we will work with the subgroup of automorphisms (P, Q) where all non-zero entries are third roots of unity, we denote this group Aut(H). Consider now the projection maps ρ (P, Q) → P and ρ (P, Q) → Q. Since √1 H is unitary, and for any automorphism 1 2 6 6 −1 (P, Q) of H the identity HQH = P holds, it follows that ρ1 and ρ2 are conjugate representations of Aut(H). Note further that ρi is a faithful representation, since Q = I forces P = I . Thus Aut(H) is isomorphic to a finite subgroup of monomial matrices of GLn(C). Furthermore, if Aut(H) contains a subgroup isomorphic to G, then the projections ρ1 and ρ2 onto the first and second components of Aut(H) give two conjugate representations of G by monomial matrices. Every monomial matrix has a unique factorisation P = DK where D is diagonal and K is a permutation matrix. The projection π : P → K is a homomorphism for any group of monomial matrices. In general, the representation ρ π ρ π Aut(H) 1 is not linearly equivalent to the representation Aut(H) 2 . As mentioned above, this phenomenon was first observed by Hall, who showed that the automorphism group of a Hadamard matrix of order 12 is isomorphic to 2 · M12, and that ρ1π and ρ2π realise the two inequivalent actions of M12 on 12 points [8]. This interpretation of the outer automorphism of M12 was also used by Elkies, Conway and Martin in their analysis of the Mathieu groupoid M13 [4]. Throughout this note we use the following shorthand for monomial matrices: we list the elements of the diagonal matrix D, and give the cycle notation for K as a permutation of the columns of the identity matrix (i.e. a right action). Consider the following pairs of monomial matrices. τ1 := ([1, 1, 1, 1, 1, 1](2, 3, 4, 5, 6), [1, 1, 1, 1, 1, 1](2, 3, 4, 5, 6)) τ2 := ([1, 1,ω,ω, ω, ω](1, 2), [1, 1, ω,ω,ω,ω] (1, 2)(3, 6)(4, 5)) . We define ∗ to be the entry-wise complex conjugation map, and consider the group X =τ1,τ2, ∗. Construction of the outer automorphism... 455 10 Proposition 1 The group X is of the form 3 · S6 · 2. τ ∗ = τ τ ∗ = τ −1 =τ ,τ = ∗ Proof Since 1 1 and 2 2 , we have that X0 1 2 is normal in X. Hence X X0 , with X0 of index 2 in X. The commutator [τ2, ∗] = ([1, 1,ω,ω, ω, ω], [1, 1, ω,ω,ω,ω]) consists of diagonal matrices; furthermore τ := [τ , ∗]−1τ = (( , ), ( , )( , )( , )), 2 2 2 1 2 1 2 3 6 4 5 a pair of permutation matrices. Recall that s, t | s6 = t2 = (st)5 =[t, s2]2 =[t, s3]2 = 1 is a presentation for S = τ 6 (see [1], for example). A computation with t 2 and = τ τ = (( , , , , , ), ( , , )( , )) s 1 2 1 2 3 4 5 6 1 2 6 3 5 , =τ ,τ shows that all the relations in this presentation hold for these elements s t, and hence Y 1 2 is isomorphic ρ π ∼ to a quotient of S6. On the other hand, Y 1 is easily seen to be isomorphic to S6, so we conclude that Y = S6. Now let N be the subgroup of X consisting of all elements for which each component is a diagonal matrix. Since ρ ρ ρ τ i and τ i have determinants in {±1}, every element of the projection X i also has determinant ±1. However all 1 2 0 ρ ρi i the elements of N have third roots of unity along the diagonal, and so must have determinant 1. As a result, X0 ∼ 5 is isomorphic to a subgroup of M S6 where M = 3 is the group of unimodular diagonal matrices with entries ρ π from ω, and S6 acts as Y i . The only non-trivial S6-submodule of M is the constant module of order 3. τ i Define ni+1 := [τ2, ∗] 1 for each i ≥ 1. (We shift subscripts because the action of τ1 on [τ2, ∗] gives elements of N which have the non-initial rows of H6 as the diagonal of the first component.) Since [τ2, ∗] ∈ X0,wehave ni ∈ X0 for 2 ≤ i ≤ 6. Observe that 2 2 = ([ , , , ,ω,ω], , , , , ω, ω ) n3n4n5 1 1 1 1 [1 1 1 1 ] τ ( 2 2) 2 = ([ , , , ,ω,ω], , ,ω,ω, , ) . n3n4n5 1 1 1 1 [1 1 1 1] ρ ρ ρ So neither of the projections N 1 , N 2 are onto the constant module, and the kernel of N 1 is neither trivial nor the constant module. It follows that N =∼ M × M. Finally, we observe that monomial matrices normalise diagonal matrices, and that X0 acts as a group of monomial matrices in each component. It follows that N X0, and that Y is a complement of N in X0. Since ∗ acts on N by inversion, N X. (P,Q) −1 The group X has a natural action on 6 × 6 matrices over C where (P, Q) ∈ X0 acts as H = P HQ, and ◦ ∗ acts by complex conjugation. We compute the stabiliser of H6 under this action. We denote this group Aut (H6) to emphasise that this is a group of semi-linear transformations in its action on the normal subgroup N. We require the subgroups X0, Y and N defined in Proposition 1 in the proof of the following.
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